Well, as far as the terminology: the alternator is the entire collection of parts, windings, coils, cores, interconnections, etc. that collectively generate alternating current (AC) which gets rectified into direct current (DC) and regulated to charge the battery and run the bike's electrical loads.
The stator, specifically, is the set of stationary windings in which the alternating current is generated by either a permanent magnet (rotating either within or around the outside of those windings) or by an electromagnet, which is a variable magnet created by varying an electric current in yet another set of windings called the field winding or just field.
As for why the system won't charge your battery, well, any of those components can be bad, including the stator, the field (if so-equipped), the interconnections, the rectifier, the regulator or the battery itself.
Search search search for a step-by-step troubleshooting guide to charging problems. They're on here somewhere...
Good luck, it's not really rocket science...
Kirk